+def is_positive_on(L,K):
+ r"""
+ Determine whether or not ``L`` is positive on ``K``.
+
+ We say that ``L`` is positive on ``K`` if `L\left\lparen x
+ \right\rparen` belongs to ``K`` for all `x` in ``K``. This
+ property need only be checked for generators of ``K``.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is positive on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is positive
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ positive on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is nonnegative.
+
+ EXAMPLES:
+
+ Positive operators on the nonnegative orthant are nonnegative
+ matrices::
+
+ sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+ sage: L = random_matrix(QQ,3).apply_map(abs)
+ sage: is_positive_on(L,K)
+ True
+
+ TESTS:
+
+ The identity is always positive in a nontrivial space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_positive_on(L,K)
+ True
+
+ As is the "zero" transformation::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_positive_on(L,K)
+ True
+
+ Everything in ``K.positive_operators_gens()`` should be
+ positive on ``K``::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_positive_on(L,K)
+ ....: for L in K.positive_operators_gens() ])
+ True
+ sage: all([ is_positive_on(L.change_ring(SR),K)
+ ....: for L in K.positive_operators_gens() ])
+ True
+
+ """
+ if L.base_ring().is_exact():
+ # This could potentially be extended to other types of ``K``...
+ return all([ L*x in K for x in K ])
+ elif L.base_ring() is SR:
+ # Fall back to inequality-checking when the entries of ``L``
+ # might be symbolic.
+ return all([ s*(L*x) >= 0 for x in K for s in K.dual() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+
+def is_cross_positive_on(L,K):
+ r"""
+ Determine whether or not ``L`` is cross-positive on ``K``.
+
+ We say that ``L`` is cross-positive on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle \ge 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. This property need only be checked for generators of
+ ``K`` and its dual.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is cross-positive
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ cross-positive on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is nonnegative.
+
+ EXAMPLES:
+
+ The identity is always cross-positive in a nontrivial space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
+ True
+
+ As is the "zero" transformation::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
+ True
+
+ TESTS:
+
+ Everything in ``K.cross_positive_operators_gens()`` should be
+ cross-positive on ``K``::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_cross_positive_on(L,K)
+ ....: for L in K.cross_positive_operators_gens() ])
+ True
+ sage: all([ is_cross_positive_on(L.change_ring(SR),K)
+ ....: for L in K.cross_positive_operators_gens() ])
+ True
+
+ """
+ if L.base_ring().is_exact() or L.base_ring() is SR:
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
+ else:
+ # The only inexact ring that we're willing to work with is SR,
+ # since it can still be exact when working with symbolic
+ # constants like pi and e.
+ raise ValueError('base ring of operator L is neither SR nor exact')
+
+
+def is_Z_on(L,K):
+ r"""
+ Determine whether or not ``L`` is a Z-operator on ``K``.
+
+ We say that ``L`` is a Z-operator on ``K`` if `\left\langle
+ L\left\lparenx\right\rparen,s\right\rangle \le 0` for all pairs
+ `\left\langle x,s \right\rangle` in the complementarity set of
+ ``K``. It is known that this property need only be
+ checked for generators of ``K`` and its dual.
+
+ A matrix is a Z-operator on ``K`` if and only if its negation is a
+ cross-positive operator on ``K``.
+
+ INPUT:
+
+ - ``L`` -- A linear transformation or matrix.
+
+ - ``K`` -- A polyhedral closed convex cone.
+
+ OUTPUT:
+
+ ``True`` if it can be proven that ``L`` is a Z-operator on ``K``,
+ and ``False`` otherwise.
+
+ .. WARNING::
+
+ If this function returns ``True``, then ``L`` is a Z-operator
+ on ``K``. However, if ``False`` is returned, that could mean one
+ of two things. The first is that ``L`` is definitely not
+ a Z-operator on ``K``. The second is more of an "I don't know"
+ answer, returned (for example) if we cannot prove that an inner
+ product is nonnegative.
+
+ EXAMPLES:
+
+ The identity is always a Z-operator in a nontrivial space::
+
+ sage: set_random_seed()
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_Z_on(L,K)
+ True
+
+ As is the "zero" transformation::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_Z_on(L,K)
+ True
+
+ TESTS:
+
+ Everything in ``K.Z_operators_gens()`` should be a Z-operator
+ on ``K``::
+
+ sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+ sage: all([ is_Z_on(L,K)
+ ....: for L in K.Z_operators_gens() ])
+ True
+ sage: all([ is_Z_on(L.change_ring(SR),K)
+ ....: for L in K.Z_operators_gens() ])
+ True
+
+ """
+ return is_cross_positive_on(-L,K)
+
+
+def is_lyapunov_like_on(L,K):